Mathbox for Stanislas Polu |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > int-ineqtransd | Structured version Visualization version GIF version |
Description: InequalityTransitivity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
Ref | Expression |
---|---|
int-ineqtransd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
int-ineqtransd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
int-ineqtransd.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
int-ineqtransd.4 | ⊢ (𝜑 → 𝐵 ≤ 𝐴) |
int-ineqtransd.5 | ⊢ (𝜑 → 𝐶 ≤ 𝐵) |
Ref | Expression |
---|---|
int-ineqtransd | ⊢ (𝜑 → 𝐶 ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | int-ineqtransd.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
2 | int-ineqtransd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | int-ineqtransd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
4 | int-ineqtransd.5 | . 2 ⊢ (𝜑 → 𝐶 ≤ 𝐵) | |
5 | int-ineqtransd.4 | . 2 ⊢ (𝜑 → 𝐵 ≤ 𝐴) | |
6 | 1, 2, 3, 4, 5 | letrd 10785 | 1 ⊢ (𝜑 → 𝐶 ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 class class class wbr 5057 ℝcr 10524 ≤ cle 10664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-pre-lttri 10599 ax-pre-lttrn 10600 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 |
This theorem is referenced by: (None) |
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